47 research outputs found
Minimal Connectivity
A k-connected graph such that deleting any edge / deleting any vertex /
contracting any edge results in a graph which is not k-connected is called
minimally / critically / contraction-critically k-connected. These three
classes play a prominent role in graph connectivity theory, and we give a brief
introduction with a light emphasis on reduction- and construction theorems for
classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page
A note on uniquely 10-colorable graphs
We prove for k at most 10, that every graph of chromatic number k with a
unique k-coloring admits a clique minor of order k
An elementary proof of Frank\u27s constructive characterization of the graphs having k edge disjoint spanning trees
Kempe Chains and Rooted Minors
A (minimal) transversal of a partition is a set which contains exactly one
element from each member of the partition and nothing else. A coloring of a
graph is a partition of its vertex set into anticliques, that is, sets of
pairwise nonadjacent vertices. We study the following problem: Given a
transversal of a proper coloring of some graph , is there
a partition of a subset of into connected sets such that
is a transversal of and such that two sets of
are adjacent if their corresponding vertices from are connected by a path
in using only two colors?
It has been conjectured by the first author that for any transversal of a
coloring of order of some graph such that any pair of
color classes induces a connected graph, there exists such a partition
with pairwise adjacent sets (which would prove Hadwiger's
conjecture for the class of uniquely optimally colorable graphs); this is open
for each , here we give a proof for the case that and the
subgraph induced by is connected. Moreover, we show that for , it
is not sufficient for the existence of as above just to force
any two transversal vertices to be connected by a 2-colored path